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arXiv 1609.09415math.AP
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变号非线性分数阶薛定谔方程的解

Solutions of the fractional Schrödinger equation with sign-changing nonlinearity

Bartosz Bieganowski

发表信息：
J. Math. Anal. Appl., Vol. 450, Issue 1, (2017), p. 461-479
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摘要：

我们寻找一个非线性分数阶薛定谔方程$$(-Δ)^{α/ 2}u + V(x)u = f(x,u)-Γ(x)|u|^{q-2}u\hbox{ on }\mathbb{R}^N,$$的解，其中$V$是强制性的，或者$V=V_{per} + V_{loc}$是$x$中的周期性势$V_{per}$和局域势$V_{loc}$的和，$Γ\in L^{\infty}(\mathbb{R}^N)$是$x$中的周期性势，$Γ(x)\geq 0$对于a.e. $x\in\mathbb{R}^N$和$2<q<2^*_α$。如果$f$具有亚临界增长，但高于$Γ(x)|u|^{q-2}u$，则我们发现基态解是Nehari流形上的极小元.此外，我们表明，如果$f$是奇数在$u$和$V$是周期的，这个方程承认无穷多个解决方案，这是两两几何不同。最后，我们得到了在强制势$V$情形下的存在性结果。

英文摘要：

We look for a solutions to a nonlinear, fractional Schrödinger equation $$(-Δ)^{α/ 2}u + V(x)u = f(x,u)-Γ(x)|u|^{q-2}u\hbox{ on }\mathbb{R}^N,$$ where potential $V$ is coercive or $V=V_{per} + V_{loc}$ is a sum of periodic in $x$ potential $V_{per}$ and localized potential $V_{loc}$, $Γ\in L^{\infty}(\mathbb{R}^N)$ is periodic in $x$, $Γ(x)\geq 0$ for a.e. $x\in\mathbb{R}^N$ and $2<q<2^*_α$. If $f$ has the subcritical growth, but higher than $Γ(x)|u|^{q-2}u$, then we find a ground state solution being a minimizer on the Nehari manifold. Moreover we show that if $f$ is odd in $u$ and $V$ is periodic, this equation admits infinitely many solutions, which are pairwise geometrically distinct. Finally, we obtain the existence result in the case of coercive potential $V$.

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